Integrand size = 11, antiderivative size = 77 \[ \int \frac {1}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {x}{7 a \left (a+b x^2\right )^{7/2}}+\frac {6 x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {8 x}{35 a^3 \left (a+b x^2\right )^{3/2}}+\frac {16 x}{35 a^4 \sqrt {a+b x^2}} \] Output:
1/7*x/a/(b*x^2+a)^(7/2)+6/35*x/a^2/(b*x^2+a)^(5/2)+8/35*x/a^3/(b*x^2+a)^(3 /2)+16/35*x/a^4/(b*x^2+a)^(1/2)
Time = 0.00 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.66 \[ \int \frac {1}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {35 a^3 x+70 a^2 b x^3+56 a b^2 x^5+16 b^3 x^7}{35 a^4 \left (a+b x^2\right )^{7/2}} \] Input:
Integrate[(a + b*x^2)^(-9/2),x]
Output:
(35*a^3*x + 70*a^2*b*x^3 + 56*a*b^2*x^5 + 16*b^3*x^7)/(35*a^4*(a + b*x^2)^ (7/2))
Time = 0.17 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.21, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {209, 209, 209, 208}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{9/2}} \, dx\) |
\(\Big \downarrow \) 209 |
\(\displaystyle \frac {6 \int \frac {1}{\left (b x^2+a\right )^{7/2}}dx}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\) |
\(\Big \downarrow \) 209 |
\(\displaystyle \frac {6 \left (\frac {4 \int \frac {1}{\left (b x^2+a\right )^{5/2}}dx}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right )}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\) |
\(\Big \downarrow \) 209 |
\(\displaystyle \frac {6 \left (\frac {4 \left (\frac {2 \int \frac {1}{\left (b x^2+a\right )^{3/2}}dx}{3 a}+\frac {x}{3 a \left (a+b x^2\right )^{3/2}}\right )}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right )}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\) |
\(\Big \downarrow \) 208 |
\(\displaystyle \frac {6 \left (\frac {4 \left (\frac {2 x}{3 a^2 \sqrt {a+b x^2}}+\frac {x}{3 a \left (a+b x^2\right )^{3/2}}\right )}{5 a}+\frac {x}{5 a \left (a+b x^2\right )^{5/2}}\right )}{7 a}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}}\) |
Input:
Int[(a + b*x^2)^(-9/2),x]
Output:
x/(7*a*(a + b*x^2)^(7/2)) + (6*(x/(5*a*(a + b*x^2)^(5/2)) + (4*(x/(3*a*(a + b*x^2)^(3/2)) + (2*x)/(3*a^2*Sqrt[a + b*x^2])))/(5*a)))/(7*a)
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
Time = 0.34 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.62
method | result | size |
gosper | \(\frac {x \left (16 b^{3} x^{6}+56 a \,b^{2} x^{4}+70 a^{2} b \,x^{2}+35 a^{3}\right )}{35 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{4}}\) | \(48\) |
trager | \(\frac {x \left (16 b^{3} x^{6}+56 a \,b^{2} x^{4}+70 a^{2} b \,x^{2}+35 a^{3}\right )}{35 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{4}}\) | \(48\) |
pseudoelliptic | \(\frac {x \left (16 b^{3} x^{6}+56 a \,b^{2} x^{4}+70 a^{2} b \,x^{2}+35 a^{3}\right )}{35 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{4}}\) | \(48\) |
orering | \(\frac {x \left (16 b^{3} x^{6}+56 a \,b^{2} x^{4}+70 a^{2} b \,x^{2}+35 a^{3}\right )}{35 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{4}}\) | \(48\) |
default | \(\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\) | \(74\) |
Input:
int(1/(b*x^2+a)^(9/2),x,method=_RETURNVERBOSE)
Output:
1/35*x*(16*b^3*x^6+56*a*b^2*x^4+70*a^2*b*x^2+35*a^3)/(b*x^2+a)^(7/2)/a^4
Time = 0.08 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left (16 \, b^{3} x^{7} + 56 \, a b^{2} x^{5} + 70 \, a^{2} b x^{3} + 35 \, a^{3} x\right )} \sqrt {b x^{2} + a}}{35 \, {\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\right )}} \] Input:
integrate(1/(b*x^2+a)^(9/2),x, algorithm="fricas")
Output:
1/35*(16*b^3*x^7 + 56*a*b^2*x^5 + 70*a^2*b*x^3 + 35*a^3*x)*sqrt(b*x^2 + a) /(a^4*b^4*x^8 + 4*a^5*b^3*x^6 + 6*a^6*b^2*x^4 + 4*a^7*b*x^2 + a^8)
Leaf count of result is larger than twice the leaf count of optimal. 1265 vs. \(2 (70) = 140\).
Time = 1.18 (sec) , antiderivative size = 1265, normalized size of antiderivative = 16.43 \[ \int \frac {1}{\left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(b*x**2+a)**(9/2),x)
Output:
35*a**14*x/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 700*a**(31/2)*b **3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a) + 210*a**(27/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqr t(1 + b*x**2/a)) + 175*a**13*b*x**3/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 210 *a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + b* x**2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4* x**8*sqrt(1 + b*x**2/a) + 210*a**(27/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 35 *a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 371*a**12*b**2*x**5/(35*a**(37 /2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a** (33/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x **2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a) + 210*a**(27/2)*b**5*x **10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 42 9*a**11*b**3*x**7/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2* sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 700*a**( 31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x* *2/a) + 210*a**(27/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x* *12*sqrt(1 + b*x**2/a)) + 286*a**10*b**4*x**9/(35*a**(37/2)*sqrt(1 + b*x** 2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*s qrt(1 + b*x**2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a*...
Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {16 \, x}{35 \, \sqrt {b x^{2} + a} a^{4}} + \frac {8 \, x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {6 \, x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} \] Input:
integrate(1/(b*x^2+a)^(9/2),x, algorithm="maxima")
Output:
16/35*x/(sqrt(b*x^2 + a)*a^4) + 8/35*x/((b*x^2 + a)^(3/2)*a^3) + 6/35*x/(( b*x^2 + a)^(5/2)*a^2) + 1/7*x/((b*x^2 + a)^(7/2)*a)
Time = 0.14 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.71 \[ \int \frac {1}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left (2 \, {\left (4 \, x^{2} {\left (\frac {2 \, b^{3} x^{2}}{a^{4}} + \frac {7 \, b^{2}}{a^{3}}\right )} + \frac {35 \, b}{a^{2}}\right )} x^{2} + \frac {35}{a}\right )} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} \] Input:
integrate(1/(b*x^2+a)^(9/2),x, algorithm="giac")
Output:
1/35*(2*(4*x^2*(2*b^3*x^2/a^4 + 7*b^2/a^3) + 35*b/a^2)*x^2 + 35/a)*x/(b*x^ 2 + a)^(7/2)
Time = 0.23 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {16\,x}{35\,a^4\,\sqrt {b\,x^2+a}}+\frac {8\,x}{35\,a^3\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {6\,x}{35\,a^2\,{\left (b\,x^2+a\right )}^{5/2}}+\frac {x}{7\,a\,{\left (b\,x^2+a\right )}^{7/2}} \] Input:
int(1/(a + b*x^2)^(9/2),x)
Output:
(16*x)/(35*a^4*(a + b*x^2)^(1/2)) + (8*x)/(35*a^3*(a + b*x^2)^(3/2)) + (6* x)/(35*a^2*(a + b*x^2)^(5/2)) + x/(7*a*(a + b*x^2)^(7/2))
Time = 0.24 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.21 \[ \int \frac {1}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {35 \sqrt {b \,x^{2}+a}\, a^{3} b x +70 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} x^{3}+56 \sqrt {b \,x^{2}+a}\, a \,b^{3} x^{5}+16 \sqrt {b \,x^{2}+a}\, b^{4} x^{7}-16 \sqrt {b}\, a^{4}-64 \sqrt {b}\, a^{3} b \,x^{2}-96 \sqrt {b}\, a^{2} b^{2} x^{4}-64 \sqrt {b}\, a \,b^{3} x^{6}-16 \sqrt {b}\, b^{4} x^{8}}{35 a^{4} b \left (b^{4} x^{8}+4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}+4 a^{3} b \,x^{2}+a^{4}\right )} \] Input:
int(1/(b*x^2+a)^(9/2),x)
Output:
(35*sqrt(a + b*x**2)*a**3*b*x + 70*sqrt(a + b*x**2)*a**2*b**2*x**3 + 56*sq rt(a + b*x**2)*a*b**3*x**5 + 16*sqrt(a + b*x**2)*b**4*x**7 - 16*sqrt(b)*a* *4 - 64*sqrt(b)*a**3*b*x**2 - 96*sqrt(b)*a**2*b**2*x**4 - 64*sqrt(b)*a*b** 3*x**6 - 16*sqrt(b)*b**4*x**8)/(35*a**4*b*(a**4 + 4*a**3*b*x**2 + 6*a**2*b **2*x**4 + 4*a*b**3*x**6 + b**4*x**8))